factor 2x^2-8x-42

{ "query": { "display": "factor $$2x^{2}-8x-42$$", "symbolab_question": "FACTOR#factor 2x^{2}-8x-42" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "2(x+3)(x-7)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Factor $$2x^{2}-8x-42:{\\quad}2\\left(x+3\\right)\\left(x-7\\right)$$", "input": "2x^{2}-8x-42", "steps": [ { "type": "interim", "title": "Factor out common term $$2:{\\quad}2\\left(x^{2}-4x-21\\right)$$", "input": "2x^{2}-8x-42", "steps": [ { "type": "step", "primary": "Rewrite $$8$$ as $$2\\cdot\\:4$$", "secondary": [ "Rewrite $$42$$ as $$2\\cdot\\:21$$" ], "result": "=2x^{2}-2\\cdot\\:4x-2\\cdot\\:21" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=2\\left(x^{2}-4x-21\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Factor Take Out Common Term Title 1Eq" } }, { "type": "step", "result": "=2\\left(x^{2}-4x-21\\right)" }, { "type": "interim", "title": "Factor $$x^{2}-4x-21:{\\quad}\\left(x+3\\right)\\left(x-7\\right)$$", "input": "x^{2}-4x-21", "steps": [ { "type": "interim", "title": "Break the expression into groups", "input": "x^{2}-4x-21", "steps": [ { "type": "definition", "title": "Definition", "text": "For $$ax^{2}+bx+c\\:$$find $$u,\\:v\\:$$ such that: $$u\\cdot\\:v=a\\cdot\\:c\\:$$and $$u+v=b$$<br/>and group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "secondary": [ "$$a=1,\\:b=-4,\\:c=-21$$", "$$u*v=-21,\\:u+v=-4$$" ] }, { "type": "interim", "title": "Factors of $$21:{\\quad}1,\\:3,\\:7,\\:21$$", "input": "21", "steps": [ { "type": "definition", "title": "Divisors (Factors)", "text": "Factors are numbers we can multiply together to get another number" }, { "type": "interim", "title": "Find the Prime factors of $$21:{\\quad}3,\\:7$$", "input": "21", "steps": [ { "type": "step", "primary": "$$21\\:$$divides by $$3\\quad\\:21=7\\cdot\\:3$$", "result": "=3\\cdot\\:7" }, { "type": "step", "primary": "$$3,\\:7$$ are all prime numbers, therefore no further factorization is possible", "result": "=3\\cdot\\:7" } ], "meta": { "interimType": "Find The Prime Factors Of Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRtEtBPmxUqyKKz2a3ulJmL+BBTEk/JQ2cZ9WKuRzClU7UyQJGJhlEk8mqOp0dvktKptlmTzO+oYqKr2Nj1dLY7IzXPlZOPbrJTojBoCz7go7" } }, { "type": "step", "primary": "Add the prime factors: ", "result": "3,\\:7" }, { "type": "step", "primary": "Add 1 and the number $$21\\:$$ itself", "result": "1,\\:21" }, { "type": "step", "primary": "The factors of $$21$$", "result": "1,\\:3,\\:7,\\:21" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Factors Top 1Eq" } }, { "type": "interim", "title": "Negative factors of $$21:{\\quad}-1,\\:-3,\\:-7,\\:-21$$", "steps": [ { "type": "step", "primary": "Multiply the factors by $$-1$$ to get the negative factors", "result": "-1,\\:-3,\\:-7,\\:-21" } ], "meta": { "interimType": "Negative Factors Top 1Eq" } }, { "type": "interim", "title": "For every two factors such that $$u*v=-21,\\:$$check if $$u+v=-4$$", "steps": [ { "type": "step", "primary": "Check $$u=1,\\:v=-21:\\quad\\:u*v=-21,\\:u+v=-20\\quad\\Rightarrow\\quad\\:$$False", "secondary": [ "Check $$u=3,\\:v=-7:\\quad\\:u*v=-21,\\:u+v=-4\\quad\\Rightarrow\\quad\\:$$True", "Check $$u=7,\\:v=-3:\\quad\\:u*v=-21,\\:u+v=4\\quad\\Rightarrow\\quad\\:$$False", "Check $$u=21,\\:v=-1:\\quad\\:u*v=-21,\\:u+v=20\\quad\\Rightarrow\\quad\\:$$False" ] } ], "meta": { "interimType": "Factor Break Into Groups Check UV Combinations 2Eq" } }, { "type": "step", "result": "u=3,\\:v=-7" }, { "type": "step", "primary": "Group into $$\\left(ax^{2}+ux\\right)+\\left(vx+c\\right)$$", "result": "\\left(x^{2}+3x\\right)+\\left(-7x-21\\right)" } ], "meta": { "interimType": "Factor Break Into Groups 0Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7mbLwQunn5/3JzwqIhKjMwqBDgWd5OmYJ6uKEvmpYkSgrnvpYQ0aygJfymuicdqpZxbXdJC0VO14XGaS2e1N17dyfY+M+XdwfX2DisUuAkI/weM8k3FbNdGF+k9z1Nfk9Bp3GEqyAv0KTmv9ly2LYFxE8uwPJldwY8ynhYzKxL5odxiol7oWpl+ok4aaIaYadn1fBl+YMO9MWjD6xr3iW9bCI2sSeA74029n2yo277ZU=" } }, { "type": "step", "result": "=\\left(x^{2}+3x\\right)+\\left(-7x-21\\right)" }, { "type": "interim", "title": "Factor out $$x\\:$$from $$x^{2}+3x:\\quad\\:x\\left(x+3\\right)$$", "input": "x^{2}+3x", "steps": [ { "type": "step", "primary": "Apply exponent rule: $$a^{b+c}=a^{b}a^{c}$$", "secondary": [ "$$x^{2}=xx$$" ], "result": "=xx+3x", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Factor out common term $$x$$", "result": "=x\\left(x+3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7fyAweyQvVwJsSQQKDjm97DAclYiY05CGstcAePg+jbh1C4k27jzuj6KLnWyn2hb8bFwMwSf2fjKgNqIbLArbNCVLOgGdTSfbpMS+IgOFGZx24affICWdjh78fVRVNyyCmi2FsF7QWDlTFS2ckOasv2H0H6aRAjgdRYKB9SVOsPKwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Factor out $$-7\\:$$from $$-7x-21:\\quad\\:-7\\left(x+3\\right)$$", "input": "-7x-21", "steps": [ { "type": "step", "primary": "Rewrite $$-21$$ as $$-7\\cdot\\:3$$", "result": "=-7x-7\\cdot\\:3" }, { "type": "step", "primary": "Factor out common term $$-7$$", "result": "=-7\\left(x+3\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Factor Out Specific 3Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7nSEJf6NykNM/HxbNHpb6ihyOUtErWNRXbTqqjj69E3uBgn/GSIMdAWH6UHJYQzHal2K3+zokN8SzKAZPagiBmEPOYwmpoyyC/tH12Kh2rHrMPF9tfONRiFlp3oxDMg6Vcv0lgHjcV/apUzqvylks59FA7BDoxxaZce2BmLVv3lwkt3WiGR7ZaCaXvz77bMjS" } }, { "type": "step", "result": "=x\\left(x+3\\right)-7\\left(x+3\\right)" }, { "type": "step", "primary": "Factor out common term $$x+3$$", "result": "=\\left(x+3\\right)\\left(x-7\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "solvingClass": "Solver2", "interimType": "Generic Factor Specific 1Eq" } }, { "type": "step", "result": "=2\\left(x+3\\right)\\left(x-7\\right)" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }